A001855 Sorting numbers: maximal number of comparisons for sorting n elements by binary insertion.

0, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, 45, 49, 54, 59, 64, 69, 74, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Equals n-1 times the expected number of probes for a successful binary search in a size n-1 list.

Piecewise linear: breakpoints at powers of 2 with values given by A000337.

a(n) is the number of digits in the binary representation of all the numbers 1 to n-1. - Hieronymus Fischer, Dec 05 2006

It is also coincidentally the maximum number of comparisons for merge sort. - Li-yao Xia, Nov 18 2015

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.3.1, Eq. (3); Sect. 6.2.1 (4).
• J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 48.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Tianxing Tao, On optimal arrangement of 12 points, pp. 229-234 in Combinatorics, Computing and Complexity, ed. D. Du and G. Hu, Kluwer, 1989.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Michael Albert, Michael Engen, Jay Pantone, and Vincent Vatter, Universal layered permutations, arXiv:1710.04240 [math.CO], (2017).
• Michael Albert, Michael Engen, Jay Pantone, and Vincent Vatter, Universal Layered Permutations, Electronic Journal of Combinatorics. Volume 25(3), 2018, #P3.23.
• J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
• Sung-Hyuk Cha, On Integer Sequences Derived from Balanced k-ary Trees, Applied Mathematics in Electrical and Computer Engineering, 2012.
• Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 36.
• Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
• D. Knuth, Letter to N. J. A. Sloane, date unknown
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
• R. Stephan, Some divide-and-conquer sequences ...
• R. Stephan, Table of generating functions
• Eric Weisstein's World of Mathematics, Sorting.
• Index entries for sequences related to sorting

Crossrefs

Partial sums of A029837.

Cf. A003071, A000337, A030190, A030308, A061168, A123753, A373709.

Programs

Haskell

import Data.List (transpose)
a001855 n = a001855_list !! n
a001855_list = 0 : zipWith (+) [1..] (zipWith (+) hs $ tail hs) where
   hs = concat $ transpose [a001855_list, a001855_list]
-- Reinhard Zumkeller, Jun 03 2013

Maple

a := proc(n) local k; k := ilog2(n) + 1; 1 + n*k - 2^k end; # N. J. A. Sloane, Dec 01 2007 [edited by Peter Luschny, Nov 30 2017]

Mathematica

a[n_?EvenQ] := a[n] = n + 2a[n/2] - 1; a[n_?OddQ] := a[n] = n + a[(n+1)/2] + a[(n-1)/2] - 1; a[1] = 0; a[2] = 1; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Nov 23 2011, after Pari *)
a[n_] := n IntegerLength[n, 2] - 2^IntegerLength[n, 2] + 1;
Table[a[n], {n, 1, 58}] (* Peter Luschny, Dec 02 2017 *)
Accumulate[BitLength[Range[0, 100]]] (* Paolo Xausa, Sep 30 2024 *)

PARI

a(n)=if(n<2,0,n-1+a(n\2)+a((n+1)\2))

PARI

a(n)=local(m);if(n<2,0,m=length(binary(n-1));n*m-2^m+1)

Python

def A001855(n):
    s, i, z = 0, n-1, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A001855(n) for n in range(1, 59)]) # Peter Luschny, Nov 30 2017

Python

def A001855(n): return n*(m:=(n-1).bit_length())-(1<Chai Wah Wu, Mar 29 2023

Formula

Let n = 2^(k-1) + g, 0 <= g <= 2^(k-1); then a(n) = 1 + n*k - 2^k. - N. J. A. Sloane, Dec 01 2007

a(n) = Sum_{k=1..n}ceiling(log_2 k) = n*ceiling(log_2 n) - 2^ceiling(log_2(n)) + 1.

a(n) = a(floor(n/2)) + a(ceiling(n/2)) + n - 1.

G.f.: x/(1-x)^2 * Sum_{k>=0} x^2^k. - Ralf Stephan, Apr 13 2002

a(1)=0, for n>1, a(n) = ceiling(n*a(n-1)/(n-1)+1). - Benoit Cloitre, Apr 26 2003

a(n) = n-1 + min { a(k)+a(n-k) : 1 <= k <= n-1 }, cf. A003314. - Vladeta Jovovic, Aug 15 2004

a(n) = A061168(n-1) + n - 1 for n>1. - Hieronymus Fischer, Dec 05 2006

a(n) = A123753(n-1) - n. - Peter Luschny, Nov 30 2017

Extensions

Additional comments from M. D. McIlroy (mcilroy(AT)dartmouth.edu)